Spatial multiplexing is a common scheme for Multiple-In-Multiple-Out (MIMO) communication systems lying on the transmission over different antennas of independent information streams.
Spatial multiplexing is conventionally combined with precoding which improves the ergodic capacity of the resulting transmission channel. FIG. 1 particularly illustrates the example of a 2×2 MIMO communication—composed of a downlink 1 and an uplink 2—between a base station 10 and a mobile equipment, such as an User Equipment.
Considering base station 10, the latter includes a channel coding block 11 providing two independent data streams to a MIMO Precoder 12 which applies a precoding precoding based on one particular matrix W selected from a codebook  of matrices known at both the base station 10 and the UE 20, before conveying such data streams over two transmit antennas 13 and 14.
On the other side, UE 20 includes two receiving antennas 23 and 24 which generate the received streams which are then forwarded to a MIMO detector performing symbols detection following channel estimation. Generally speaking, the detection may be based on either a linear near-ML detector, as illustrated in FIG. 2, or a non-linear near-ML (near Maximum-Likehood) detection such as Sphere Decoders.
In the particular case of a closed-loop system, the detector is associated with Precoding Matrix Index block 22 which is arranged to compute one particular value of an index which is conveyed to the transmitting base station 10 through uplink 2 so that the latter can uniquely selects one advantageous matrix W to be used in the precoding process.
At this point, the closed-loop system model reads simultaneously:y=HWx+n=Hpx+n={tilde over (H)}pz+n 
with x being the transmit symbol vector withdrawn from a finite and known constellation, y being the receive symbol vector, H being the channel matrix, W being the precoding matrix and n being an additive White gaussian noise (AWGN) of varience σ2n. The effective channel matrix reads HW.
The concept of precoding in spatial multiplexing systems raises two main issues:
The first issue—which is only mentioned here for the sake of clarity and which will not be considered in the present application—consists in the design the set  given a chosen selection criterion. Such issue is particularly addressed in the following prior art reference (3):
The second issue—which is particularly addressed in the present application—consists in the determination of an advantageous precoding matrix from a codebook  through an appropriate selection criterion.
One conventional technique is known in the art and designated as the so-called Capacity-Selection Criterion (C-SC), which is recommended by 3GPP is described in reference 3GPP TS 36.211 V9.1.0 (2010-03), Technical Specification, 3rd Generation Partnership Project; Technical Specification Group Radio Access Network; Evolved Universal Terrestrial Radio Access (E-UTRA); Physical Channels and Modulation, (Release 9).
The conventional C-SC technique is particularly adapted to the use of a Minimum Mean Square Estimation (MMSE) detector and clearly optimal for such MMSE detector.
Basically, in such C-SC prior art technique, the index selection is based on the SINR maximization with the knowledge of any precoding matrix and the estimation of the channel matrix in reception, and reads:
                    SINR        k            ⁡              (                  W          ,          H                )              =                  1                              (                          I              +                                                1                                      σ                    n                    2                                                  ⁢                                  P                  H                                ⁢                                  H                  H                                ⁢                H                ⁢                                                                  ⁢                P                                      )                                k            ,            k                                -            1                              -      1        ,where ν2n is the noise variance, and under the assumption of normalized power constellations.
In the particular case of a 2×2 MIMO system, the expression above rewrites—for the first layer:
                    SINR        0            ⁡              (                  W          ,          H                )              =                                                  U            00                                    2                                                                    U              01                                            2                +                              σ            n            2                    ⁢                                                                  B                00                                                    2                          +                              σ            n            2                    ⁢                                                                  B                01                                                    2                                ,whereU=(HPHHP+σn2I)−1HPHHP 
The Capacity-Selection Criterion (C-SC) consists in choosing w such that:W=arg maxwi∈w{ΣkSINRk(Wi,H)}.
While the C-SC technique is the optimal in the case of the widely spread linear MMSE detector,—because of the maximization of the capacity which results—it should be noticed that such technique is less advantageous in the case of a more elaborated and sophisticated decoding technique, such as based on a non linear technique, such as achieved by a near-Maximum Likehood detection.
Indeed, thanks to its level of sophistication, a near-Maximum Likehood technique already achieves a high degree of capacity given a channel and, in such situation, the known Capacity-Selection Criterion (C-SC) index selection which is recommended by the 3GPP shows limited interest.
The technical problem to be solved by the present invention is to design a new index selection mechanism which is more adapted to the high efficiency of a near-ML decoder, such as a decoder using a Lattice Reduction technique.